These \(\phi \) -divergences have the advantage of simplicity, since they only perform pointwise comparisons of the measures at each position. This however totally ignores the metric of the underlying space.
It makes these divergences incapable of comparing discrete distributions with continuous ones (for instance the KL divergence is always \(+\infty \)  in these cases).
This issue motivates the use of other losses which leverage the underlying metric.
Examples of such losses are Maximum Mean Discrepancies [1]} and Optimal Transport distances [2]}, which are detailed in Sections REF  and REF .
This review focuses on Optimal Transport and its extensions, with an emphasis on their applicability for data sciences.
